I like tea. I like my teas in a particular way (plural intended). What I will now expose to you is a result of deep meditation on this subject, conducted while sipping tea at various occasions in the past three years since I was introduced to the idea that tea is not just warm, brown water (by my girlfriend, who is the expert between the two of us). No, I'm kidding. You can find this information from Wikipedia and with Google.

I subscribe to the idea that there are few wrong ways and many right ways. Since tea is an edible (or in this case, drinkable) substance, and since there exists at least one gene that affects your perception of bitterness, there are no doubt more genes involved in the sense of taste, and thus everyone will taste things slightly differently. However, I have a few generic remarks:

Clean water. This is essential and probably the most important component after tea leaf quality in the taste of the tea.

Water temperature: this depends on what kind of tea you drink. Tea brews best in hot water, but you can't use too hot since it will be bitter. The numbers here try to minimize bitterness and maximize taste.

For white tea, never use hotter than 80°C. This has a huge difference in taste, as brewing white in too hot water makes it unpleasantly bitter.

For green tea, never use hotter than 80°C or so. Again, you'll notice the bitterness.

For oolong, 90°C seems to be the best.

For black, 100°C is just fine.

How do you know, if using a kettle, when the water is of the right temperature? When water boils, there are three observable "stages" (you'll find this list with Google easily, and the origin seems to be a Chinese legend): when you get lots of small bubbles, the temperature is getting closer to 80°C; when you get long strings of small bubbles from the bottom to the top, 90°C; and huge bubbles when the water boils at 100°C.

Brewing time: this again varies with the type of tea. The more you brew, the more you'll get the bitter chemicals in the tea. As far as I remember, the bitterness of tea does not start to increase until after the first minute of seeping or so. The instructions on the bag are usually good enough; I generally vary the time and figure out the right one for myself (how I like it).

Leaf type: loose leaf or bagged. If you only can, go for loose leaf tea. Not only will your tea taste better, you will also be able to get more tea out of them. Bagged tea is lower quality, containing not only the "effective" parts of the leaf but bits and pieces of the central stalk (whatever it is called). This doesn't matter if you only use the bag once; the taste will be the same. However, with high quality loose leaf, you can reuse the leaves 4-6 times. Quite often the best brew will be the second one. To prepare loose leaf, take a look at the other replies in this node.

Tea type (oxidization): There are four main types: white, green, oolong, and black, in the order of how much they have been oxidized. Black has the strongest and fullest taste, and white the most delicate. You can also vary the type on the axes of when the leaves where picked, if they have essential oils added, etc.

That should get you started. Wikipedia is another nice starting point. As for what to put into tea or eat with tea, there are many different traditions about it. Most kinds of black are good with or without milk. I don't recommend putting anything in green, white or oolong; their taste is too delicate for milk, sugar, or honey (but do try it out if you like). Often, with green, you serve something sweet (though not excessively sweet).

The best method is to try things out for yourself, but this requires some reading at first, and preferably a high quality teashop where you can ask about teas before buying. If you're used to bagged black, I recommend going to your local supermarket and shop for loose leaf black (Twinings is not a bad choice), and experiment with it at first.